In the diaspora, where two days of yomim tovim are celebrated, the second day is typically an exact repetition of the first day. We say the same prayers, make a second Seder on Pesach, make kiddush in the Sukkah on Sukkot, etc. Shmini Atzeret appears to be an exception to this rule. The first day of Shmini Atzeret is celebrated as a semi-Sukkot holiday (it is also the second 7th day of Sukkot), and the second day is Simchat Torah, with hakafot, dancing, singing, and a variety of rituals done only on this day. Why don't we celebrate two days of Simchat Torah?

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The Bnei Bavel would finish the Torah every year by dividing the Parshios the way we do now, however the Bnei Eretz Yisroel would only finish it every 3 1/2 years and thus would not even celebrate Simchas Torah on a yearly basis. The Minhag if the Bnei Bavel was to celebrate the completion of the Torah on the 9th day of Yom Tov - on the day of Safeik Yom of Shemini Atzeres. This does not have to do with Shemini Atzeres, it has to do with the completion of the Torah. Since in Eretz Yisroel there is no Yom Tov on this day therefore they celebrate it on Shemini Atzeres.